Whakaoti i te y-3x=2text{and}7x-2y=8

$$y - 3 x = 2 \text { and } 7 x - 2 y = 8$$

Whakaoti mō y, x

$$x=12$$

$$y=38$$

Ngā Upane e Whakamahi ana i te Whakakapinga

Hei whakaoti i ētahi whārite takirua mā te whakakapinga, me whakaoti tētahi whārite i te tuatahi mō tētahi o ngā taurangi. Ka whakakapi i te otinga mō taua taurangi ki tērā o ngā whārite.

$$y-3x=2,-2y+7x=8$$

Kōwhiria tētahi o ngā whārite ka whakaotia mō te y mā te wehe i te y i te taha mauī o te tohu ōrite.

$$y-3x=2$$

Me tāpiri 3x ki ngā taha e rua o te whārite.

$$y=3x+2$$

Whakakapia te 3x+2 mō te y ki tērā atu whārite, -2y+7x=8.

$$-2\left(3x+2\right)+7x=8$$

Whakareatia -2 ki te 3x+2.

$$-6x-4+7x=8$$

Tāpiri -6x ki te 7x.

$$x-4=8$$

Me tāpiri 4 ki ngā taha e rua o te whārite.

$$x=12$$

Whakaurua te 12 mō x ki y=3x+2. I te mea kotahi anake te taurangi kei te whārite i puta, ka taea e koe te whakaoti mō y hāngai tonu.

$$y=3\times 12+2$$

Whakareatia 3 ki te 12.

$$y=36+2$$

Tāpiri 2 ki te 36.

$$y=38$$

Kua oti te pūnaha te whakatau.

$$y=38,x=12$$

Ngā Upane e Whakamahi ana i Ngā Poukapa

Tuhia ngā whārite ki te tānga ngahuru ka whakamahi i ngā poukapa hei whakaoti i te pūnaha o ngā whārite.

$$y-3x=2,-2y+7x=8$$

Tuhia ngā whārite ki te tikanga tātai poukapa.

$$\left(\begin{matrix}1&-3\\-2&7\end{matrix}\right)\left(\begin{matrix}y\\x\end{matrix}\right)=\left(\begin{matrix}2\\8\end{matrix}\right)$$

Whakarea mauī i te whārite ki te poukapa kōaro o \left(\begin{matrix}1&-3\\-2&7\end{matrix}\right).

$$inverse(\left(\begin{matrix}1&-3\\-2&7\end{matrix}\right))\left(\begin{matrix}1&-3\\-2&7\end{matrix}\right)\left(\begin{matrix}y\\x\end{matrix}\right)=inverse(\left(\begin{matrix}1&-3\\-2&7\end{matrix}\right))\left(\begin{matrix}2\\8\end{matrix}\right)$$

Ko te hua o tētahi poukapa me te kōaro ko te poukapa tuakiri.

$$\left(\begin{matrix}1&0\\0&1\end{matrix}\right)\left(\begin{matrix}y\\x\end{matrix}\right)=inverse(\left(\begin{matrix}1&-3\\-2&7\end{matrix}\right))\left(\begin{matrix}2\\8\end{matrix}\right)$$

Whakareatia ngā poukapa kei te taha mauī o te tohu ōrite.

$$\left(\begin{matrix}y\\x\end{matrix}\right)=inverse(\left(\begin{matrix}1&-3\\-2&7\end{matrix}\right))\left(\begin{matrix}2\\8\end{matrix}\right)$$

Mō te poukapa 2\times 2 \left(\begin{matrix}a&b\\c&d\end{matrix}\right), ko te \left(\begin{matrix}\frac{d}{ad-bc}&\frac{-b}{ad-bc}\\\frac{-c}{ad-bc}&\frac{a}{ad-bc}\end{matrix}\right) te poukapa kōaro, nō reira ka taea te tuhi anō te whārite poukapa hei rapanga whakarea poukapa.

$$\left(\begin{matrix}y\\x\end{matrix}\right)=\left(\begin{matrix}\frac{7}{7-\left(-3\left(-2\right)\right)}&-\frac{-3}{7-\left(-3\left(-2\right)\right)}\\-\frac{-2}{7-\left(-3\left(-2\right)\right)}&\frac{1}{7-\left(-3\left(-2\right)\right)}\end{matrix}\right)\left(\begin{matrix}2\\8\end{matrix}\right)$$

Mahia ngā tātaitanga.

$$\left(\begin{matrix}y\\x\end{matrix}\right)=\left(\begin{matrix}7&3\\2&1\end{matrix}\right)\left(\begin{matrix}2\\8\end{matrix}\right)$$

Whakareatia ngā poukapa.

$$\left(\begin{matrix}y\\x\end{matrix}\right)=\left(\begin{matrix}7\times 2+3\times 8\\2\times 2+8\end{matrix}\right)$$

Mahia ngā tātaitanga.

$$\left(\begin{matrix}y\\x\end{matrix}\right)=\left(\begin{matrix}38\\12\end{matrix}\right)$$

Tangohia ngā huānga poukapa y me x.

$$y=38,x=12$$

Ngā Upane e Whakamahi i te Tangohanga

Hei whakaoti mā te tangohanga, ko ngā tau whakarea o tētahi o ngā taurangi me mātua ōrite i ngā whārite e rua kia whakakorehia ai te taurangi ina tangohia tētahi whārite mai i tētahi atu.

$$y-3x=2,-2y+7x=8$$

Kia ōrite ai a y me -2y, whakareatia ngā kīanga tau katoa kei ia taha o te whārite tuatahi ki te -2 me ngā kīanga tau katoa kei ia taha o te whārite tuarua ki te 1.

$$-2y-2\left(-3\right)x=-2\times 2,-2y+7x=8$$

Whakarūnātia.

$$-2y+6x=-4,-2y+7x=8$$

Me tango -2y+7x=8 mai i -2y+6x=-4 mā te tango i ngā kīanga tau ōrite i ia taha o te tohu ōrite.

$$-2y+2y+6x-7x=-4-8$$

Tāpiri -2y ki te 2y. Ka whakakore atu ngā kupu -2y me 2y, ka toe he whārite me tētahi taurangi kotahi ka taea te whakaoti.

$$6x-7x=-4-8$$

Tāpiri 6x ki te -7x.

$$-x=-4-8$$

Tāpiri -4 ki te -8.

$$-x=-12$$

Whakawehea ngā taha e rua ki te -1.

$$x=12$$

Whakaurua te 12 mō x ki -2y+7x=8. I te mea kotahi anake te taurangi kei te whārite i puta, ka taea e koe te whakaoti mō y hāngai tonu.

$$-2y+7\times 12=8$$

Whakareatia 7 ki te 12.

$$-2y+84=8$$

Me tango 84 mai i ngā taha e rua o te whārite.

$$-2y=-76$$

Whakawehea ngā taha e rua ki te -2.

$$y=38$$

Kua oti te pūnaha te whakatau.

$$y=38,x=12$$